To determine the temperature to which the steel plate must be heated in order for the 1cm diameter cylinder to fit into the 0.9997cm diameter hole at 30 degrees Celsius, we can use the formula for thermal expansion.
The thermal expansion of a material, such as steel, can be calculated using the formula:
ΔL = α * L * ΔT
Where:
ΔL is the change in length or diameter,
α is the coefficient of linear expansion for the material,
L is the original length or diameter,
ΔT is the change in temperature.
For steel, the average coefficient of linear expansion is approximately 12 × 10^(-6) per degree Celsius.
In this case, the change in diameter of the hole is ΔD = 1cm - 0.9997cm = 0.0003cm.
Now, let's calculate the change in temperature (ΔT).
ΔL = α * L * ΔT
0.0003cm = (12 × 10^(-6) / °C) * 0.9997cm * ΔT
Solving for ΔT:
ΔT = 0.0003cm / (12 × 10^(-6) / °C) * 0.9997cm
ΔT ≈ 0.0125°C
Therefore, the steel plate must be heated by approximately 0.0125 degrees Celsius in order for the cylinder to fit into the hole.